-------- Original Message -------- Subject: Re: Canonical variates from first PCs of GPA residuals Date: Wed, 11 Feb 2009 07:12:43 -0800 (PST) From: andrea cardini <[email protected]> To: [email protected] Dear Peter, please, read my answers below. At 09:15 11/02/2009 -0500, you wrote:
-------- Original Message -------- Subject: Canonical variates from first PCs of GPA residuals Date: Tue, 10 Feb 2009 05:15:05 -0800 (PST) From: Peter Taylor <[email protected]> To: <[email protected]> Dear Morphometricians I am working with data where the number of landmarks (from rodent skulls) exceeds the smallest sample sizes of my groups. To circumvent statistical problems with null determinants when using canonical analysis (CVA) of the weights matrix from GPA, is it permissable to conduct CVA on the first few PCs from a PCA of the residuals, or aligned coordinates after least squares, GPA?
PCA is often used to summarize variation in a sample. Whether this summary is adequate, it's a different matter and there's a number of way to at least try to be more objective in the choice of how many PCs to include (see below). There's a number of papers where first PCs were selected to reduce dimensionality in further analyses including DA (CVA) and related techniques. This includes quite a few of my own ones (follow the links in my electronic signature - e.g., the paper on divergence in guenons and several other ones). Most of the time I tend to exclude smallest samples from these analyses. You can avoid some of the issues with dimensionality vs sample sizes by using resampling stats. For instance, you could do a series of pairwise comparisons between your samples and then use a correction to reduce the risk of inflating type I errors in multiple comparisons (e.g., sequential Bonferroni and similar ones). Again, you may find examples in some of my papers (probably the one on the Vancouver marmot published in J Evol Biol, the one on the Zanzibar red colobus which has just been published in the International Journal of Primatology and a few others already out or in press). If so how does one objectively
decide how many PCs to include, should this number be less than the smallest group sample size, or should it depend on a certain threshold of cumulative explained variance (70%) or on the eigenvalues (>1?), or on the degree of separation of groups?
You'll find plenty of stuff about this in the literature. The problem is not special of geometric morphometrics. Probably the approach I like the most for its simplicity is the one I first read in a paper by Fadda & Corti published in a special issue of Hystrix 2000. It should be avaialable online and possibly there's a link in the SUNY Morphometrics website (not sure about this). The same method is described in several papers of mine (again, those on guenons and vervets etc.): "The number of principal components to be analysed was selected by measuring the correlation between the matrix of Procrustes shape distances in the full shape space and pairwise Euclidean distances in the reduced shape space (5, 10, 15 principal components, and so on). Plots of correlation coefficients onto the number of components can be used in a similar way to scree plots to select how many variables summarize most shape variation." Also, is this approach
equivalent, or preferable, to conducting CVA on the first few relative warps from a relative warps analysis (PCA of weights matrix). I have seen both approaches in the literature but not sure which is best.
Except in a very special case, a relative warp analysis (PCA of partial warps and uniform components) is exactely the same as a PCA of Procrustes shape coordinates after projection in the tangent space. Good luck! Cheers Andrea
Many thanks Peter Dr Peter John Taylor Curator of Mammals Durban Natural Science Museum Ethekwini Libraries & Heritage P O Box 4085 Durban 4000 --------------------------------------------------------------------------- Physical address: First Floor, City Hall, Smith Street Entrance, 4001 & Research Centre, 151 Old Fort Road (cnr Wyatt St) â------------------------------------------------------------------------- Tel: + 27 31 3054162/4/5/7 Cell: 083 7924810 Fax: + 27 31 311 2242 Email: [email protected] <mailto:[email protected]> or (home): [email protected] <mailto:[email protected]> or: [email protected] <mailto:[email protected]> Internet: www.durban.gov.za/naturalscience/ <http://www.durban.gov.za/naturalscience/> -- Replies will be sent to the list. For more information visit http://www.morphometrics.org
Dr. Andrea Cardini Lecturer in Animal Biology Museo di Paleobiologia e dell'Orto Botanico, Universitá di Modena e Reggio Emilia via Università 4, 41100, Modena, Italy tel: 0039 059 2056532; fax: 0039 059 2056535 Honorary Fellow Functional Morphology and Evolution Unit, Hull York Medical School University of Hull, Cottingham Road, Hull, HU6 7RX, UK University of York, Heslington, York YO10 5DD, UK E-mail address: [email protected], [email protected], [email protected] http://hyms.fme.googlepages.com/drandreacardini http://ads.ahds.ac.uk/catalogue/archive/cerco_lt_2007/overview.cfm#metadata More on publications at: http://www.cons-dev.org/marm/MARM/EMARM/framarm/framarm.html CLICK ON THE LETTER C AND LOOK FOR "CARDINI" http://hyms.fme.googlepages.com/dr.sarahelton-publications LOOK FOR "CARDINI" -- Replies will be sent to the list. For more information visit http://www.morphometrics.org
